Features of Gaussian distribution curve

Represented To: Dr. IU khan

Represented By: Farzeen Javaid 0188-BH-CHEM-11

Course Title: Separation Techniques (Compulsory)

Course Code: CHEM-4201

Govt. College University Lahore

DEFINITON1) The area under a normal curve has a normal distribution

(a.k.a., Gaussian distribution)

2) The normal probability distribution (Gaussian distribution) is a continuous distribution which is regarded by many as the most significant probability distribution in statistics particularly in the field of statistical inference.

Symbols Used “z” – z-scores or the standard scores. The table that transforms

every normal distribution to a distribution with mean 0 and standard deviation 1. This distribution is called the standard normal distribution or simply standard distribution and the individual values are called standard scores or the z-scores.

“µ” – the Greek letter “mu,” which is the Mean, and “σ” – the Greek letter “sigma,” which is the Standard

Deviation

CHARACTERISTICS The normal curve is bell-shaped and has a single peak at the

exact center of the distribution. The arithmetic mean, median, and mode of the distribution

are equal and located at the peak. Half the area under the curve is above the peak, and the

other half is below it. The normal distribution is symmetrical about its mean. The normal distribution is asymptotic - the curve gets closer

and closer to the x-axis but never actually touches it. Unimodal - a probability distribution is said to be normal if the

mean, median and mode coincide at a single point Extends to +/- infinity- left and right tails are asymptotic with

respect to the horizontal lines Area under the curve = 1

Characteristics of a Normal Distribution

THE NORMAL DISTRIBUTION OF MATHEMATICAL FUNCTION (pdf)

The Normal Distribution

The normal curve is not a single curve but a family of curves, each of which is determined by its mean and standard deviation.

PARAMETER The normal distribution can be completely specified by two

parameters:

1. Mean

2. Standard deviation

If the mean and standard deviation are known, then one essentially knows as much as if one had access to every point in the data set.

Mean (µ) Mean : Measure of Central tendency Center or middle of data set around which observations are

lying Assuming : frequency in each class is uniformly distributed

and representable by mid point Mean for grouped data is given by

x̄�

where

n = no of observations

fi = frequency of each (ith) class interval

xi = mid point of each class interval

Standard Deviation (σ)

Standard Deviation : Measure of Dispersion Average deviation of observations around the mean Compactness or variation of data SD = root mean square deviation SD = variance = (x̄i – x̄� )²

where

n = no of observations

x> = mean of the frequency distribution

xi = mid point of each class interval

Curves with different means, same standard deviation

Curves with different means, different standard deviations

PROPERTIES OF A NORMAL DISTRIBUTION

As the curve ex̄tends farther and farther away from the mean, it gets closer and closer to the x-ax̄is but never touches it.

The points at which the curvature changes are called inflection points. The graph curves downward between the inflection points and curves upward past the inflection points to the left and to the right.

Properties of normal curve

1. The function f(x) defining the normal distribution is a proper p.d.f., i.e. f(x)≥0 and the total area under the normal curve is unity.

2. The mean and variance of the normal distribution are µ and σ2 respectively.

3. The median and the mode of the normal distribution are each equal to µ, the mean of the distribution.

4. The mean deviation of the normal distribution is approximately 4/5 of its standard deviation.

5. The normal curve has points of infection whih are equidistant from the mean.

6. For the normal distribution, the odd order moments about the mean are all zero.

Properties of normal curve7. if X is N (µ, σ2 ) and if Y a+bx, then Y is N (a+bµ, b2 σ2 ).

8. The sum of independent normal variables is a normal variable. Stated differently, if X1 is N (µ1, σ1

2) and X2 is N (µ1, σ2

2), then for independent X1 and X2, X1 + X2 is N (µ1 + µ2, σ1

2 + σ22 ).

9. The normal curve approaches, but never really touches the horizontal axis on either side of the mean towards plus and minus infinity, that is the curve is asymptotic to the horizontal axis as x ±∞

10. The height of any normal curve is maximized at x = µ.

11. All normal curves are positive for all x. That is, f(x) > 0 for all x.

12.  The area under an entire normal curve is 1.

Properties of normal curve13. No matter what the value of µ and σ are, area under normal curve remain in certain fixed proportions within a specified number of standard deviation on either side of µ. For example the interval

µ ± σ will always contain 68.26% µ ± 2σ will always contain 95.44% µ ± 3σ will always contain 99.73%

References C. M Sher. Introduction To Statistical Theory part 1, 8th

Edition. Markazi Kutab Khana Publisher, 2002 http://www.unt.edu/rss/class/Jon/ISSS_SC/Module004/

isss_m4_normal/node4.html https://onlinecourses.science.psu.edu/stat414/node/149 https://www3.nd.edu/~rwilliam/stats1/x21.pdf